House Republicans proposed legislation Tuesday that would amend federal law to make it easier to deport Central American minors and devote $659 million in emergency funding to address the growing humanitarian crisis on the southern border, as GOP leaders try to win enough support from skeptical conservatives to pass a bill by Thursday.

The amount is far lower than the $3.7 billion request from the Obama administration, which says the money is necessary to help stem a growing influx of unaccompanied families and children entering the country illegally. And the proposed change to the 2008 anti-trafficking law is likely to be met with strong opposition from Democratic lawmakers, making it highly uncertain whether Congress can resolve its disagreements before the House breaks for a five-week summer recess by the end of the week.

"I think there is sufficient support in the House," Speaker John A. Boehner (R) told reporters after the weekly meeting of the House Republican Conference, adding a caveat: "We have a little more work to do."

The House GOP last week proposed spending $1.5 billion on the border crisis, then reduced it to $1 billion, before trimming again this week to win more support. Under the House bill filed Tuesday, $405 million would go toward increased law enforcement efforts to process the children in immigration courts, with just $197 million to go toward providing shelter and care for them.

Obama has sought $1.6 million for law enforcement efforts and $1.8 billion for shelter and care.

Even as Boehner expressed cautious optimism, the clock is working against lawmakers. They've set Thursday as departure day for a five-week break for lawmakers to head home and begin campaigning for the November elections.

At this point, Boehner believes that he has a majority of Republicans in favor of the new plan, but with no expectation of Democratic votes it's unclear if the new leadership team can get a House majority to pass the bill entirely from the GOP side of the aisle. He declined to guarantee a vote before Thursday's departure.

"I think we should do something before we go home, and we're working to get there," he said.

Even if the House acts to pass a bill, there's little sense on Capitol Hill that its plan would pass the Senate. Democrats in the upper chamber have proposed a $2.7 billion plan that has also been met with bipartisan skepticism.

Further complicating matters is that the GOP is calling for amendments to a 2008 anti-trafficking law that currently provides greater legal protections to unaccompanied children who enter the United States illegally from countries other than Mexico or Canada.

The House plan would treat all children the same as those from neighboring countries, and require that they have their immigration hearings processed within one week of being apprehended by Border Patrol. Currently, massive backlogs have created wait times of more than a year for many of the children, who are placed with relatives or in shelters as they await their hearings.

The White House had tacitly endorsed a change in recent weeks, saying it would give Homeland Security Secretary Jeh Johnson more authority to return the children to Central America. But Democrats and immigration advocates have opposed overturning the legal protections meant to protect children from sex trafficking.

On Tuesday, White House press secretary Josh Earnest objected to the House plan, arguing that the changes to the 2008 law would actually risk making things worse by imposing an "arbitrary" deadline on the deportation hearings of one week. That could force the administration to take resources from other immigration enforcement priorities and devote them to accelerating the hearings, he said.

"That sort of inflexible approach only risks bottling up the system further," Earnest said.

More than 57,000 unaccompanied minors and another 55,000 parents with children have been apprehended on the border since last October, a sharp increase from last year that has overwhelmed border patrol stations. Administration officials said the families are fleeing violence and poverty in Guatemala, El Salvador and Honduras, where most are coming from. But GOP critics say the administration's immigration policies are too lenient and have contributed to the crisis.

The Obama administration says the emergency funding will help provide additional immigration judges and asylum lawyers to speed up deportation hearings and provide more shelter space for the families awaiting their court hearings.

At a House Judiciary Committee hearing on immigration Tuesday, Chairman Bob Goodlatte (R-Va.) said: "If President Obama took seriously his duty to secure the U.S. border... there would be no crisis."